The Natural Probability of a No-Hitter

Some guys have all the luck.

Not a favorable umpire.

A lot of variables need to cooperate for a Major League Pitcher to throw a successful complete game no-hitter. The pitcher needs both good stuff and good command - a combination that many can't put together for one inning, much less one game. He needs rangey, sure-handed fielders backing him up and he needs a favorable umpire. He needs cooperative batters that expand their zone just a little more than normal. But maybe most importantly, he needs a lot of luck.

It's been said that the best baseball players in the history of the game fail 70% of the time. While not entirely true because it discounts the importance of silly little things like walks and sacrifice flies, it is the case that a very good hitter only gets to stand safely at a base after putting the ball into play in 3 out of 10 at bats. A 30% success rate when putting the ball into play gets you into the Hall of Fame, according to Ted Williams. If it's so unlikely that a player gets a hit in a given at-bat, why do so few pitchers throw no-hitters?

Batting average is not a predictive statistic; it describes only what has been done before and not what will happen now or in the future. After a number of years, though, trends do start to emerge: some guys consistently hit .280 and some poor souls consistently hit .240. And one guy hit .200 so consistently that a 20% success rate was named after him. If the best predictor of future behavior is past behavior, then batting average can be extrapolated to the long-term future and be a fairly accurate gauge of what is to come. Remember, a single game is such a small sample size that season and career averages mean nothing within the confines of nine innings.

So then, it is naturally impossible to estimate the likelihood of a no-hitter in a specific game. It is, however, possible to give a relatively accurate probability of a specific team not getting a single hit in a randomly chosen game, as long as that team features a lineup of nine players with enough history for their batting averages to be accurate. That is: if the sample size is large enough for every batter on a chosen team, we can make an educated guess as to how often an opposing pitcher will toss a no-hitter. Data is available for every pitcher's batting average against as well, so we can even see how often a given pitcher should be throwing a no-no. And finally, for two teams that play each other often - like, for example, divisional opponents - batters will have enough of a sample size against a specific pitcher (and that pitcher a batting average against for the opposing batters) so that we will be able to estimate how likely it is that a specific pitcher tosses a no-hitter against a specific team.

Tough-luck winner Yu Darvish and the Houston Astros will be the subject of this study after missing out on the opportunity to be on opposite sides of two no-hit bids, both perfect game bids through at least six innings, in 2013. Darvish and the Rangers still won both games, but they'd probably love to have added their names to the history book too.

Position	Player	Career AVG	2013 AVG
C	Jason Castro	0.255	0.276
1B	Chris Carter	0.220	0.223
2B	Jose Altuve	0.285	0.283
3B	Matt Dominguez	0.248	0.241
SS	Jonathan Villar	0.243	0.243
LF	J.D. Martinez	0.251	0.250
CF	Brandon Barnes	0.233	0.240
RF	L.J. Hoes	0.281	0.287
DH	Carlos Pena	0.233	0.209
Source: ESPN, Baseball-Reference.com

The 2013 Astros used a number of different batting lineups: 19 different players had over 100 plate attempts and 25 players (not including pitchers) registered at least 1 plate attempt. The above lineup is one common combination of many possibilities, according to most games started at each position. Carlos Pena started the second-most games at DH, behind Chris Carter, who held the top spot at both 1B and DH. Carter started more games at 1B than DH. Since he can't occupy two lineup spots, he'll be our 1B and Pena will serve as the DH. Note that, of the nine listed players, only Carlos Pena has played at the MLB level for more than four years. Most of the averages shown are likely volatile and subject to significant change as these players develop.

To determine probability of not getting a hit, we need only to subtract the above averages from 1: NoHitAVG= 1 - p_hit where p_hit is given by AVG. Each individual Astro in this lineup, in 2013 and over their careers, is likely to not get a hit in at least 71.3% of randomly selected plate attempts. It is important to remember that not every non-hit is an out. A player can still reach base on an error, a walk, or a hit-by-pitch. Since none of these breaks up a no-hitter, and since each can theoretically happen an infinite number of times during a no-hitter, the probability of a no-hitter as a result of random chance will be presented as a maximum.

The 2013 Astros with the given lineup posted a 0.041% chance of not getting a hit in a single, randomly selected game during the 2013 season. This is an improvement on the likelihood of being no-hit based on career averages. With a more established team, it might be said that the lineup had over-performed and was due to regress. Because the Astros were so young and with very little history at the Major League-level, it's likely that 2013 represents the low-water mark and that future iterations of the Astros franchise will be less likely to be no-hit in the coming years. Something to look forward to in Houston, I guess.

According to this probability, there should have been a maximum of 0.067 games in which the Astros were no-hit during the 2013 season. On average, we would expect this lineup to be no-hit once at least every 2,418 games. In more meaningful terms: this Astros lineup, if it performs at exactly this rate every year, based strictly on chance, should be no-hit once every 15 years. Yu Darvish was almost successful in doing so twice in 162 games.

* * *

Yu Darvish is a 27-year-old pitching phenom. Just now entering his physical prime, Darvish has two spectacular years in Major League Baseball after a few years of dominating Japanese hitters. In 2013, his ERA sat at 2.83 (4th), he pitched 209.1 innings, threw 277 strikeouts (1st), and had a WHIP of 1.073. The guy throws five very different pitches. He's a two-time All Star with serious Rookie of the year and Cy Young considerations.

The Rangers pitcher also holds an impressively low .207 career batting average against. In 2012, his batting average against was .220: 2012 Darvish turned hitters into Ryan Flaherty. In 2013, Darvish dropped his average against to just .194, below the Mendoza line. With just two years in the league, it's difficult to say whether his 2012 or 2013 is the outlier and which is the true average. Either way, Darvish projects to be a top-tier pitcher for years to come.

In 2013, a randomly selected at-bat against Yu Darvish had an 80.6% of being an out. The probability of Darvish throwing a no-hitter, then, is 0.296% in a randomly selected game. This is not a particularly high probability. Darvish is more likely to throw a no-hitter than you are to bowl a 300 (0.01%), date a supermodel (0.114%), or hit a hole in one (0.020%), though. Darvish should toss a no-hitter every 337.8 games started. Darvish started 29 games in 2012 and 32 games in 2013, so we can safely say that, barring injury, he pitches about 30 games every year. Therefore, Darvish should throw a no-hitter once every 11.26 years. And Yu Darvish should throw about 0.1 no-hitters in a regular 30-start season.

Interestingly, Darvish is more likely to throw a no-hitter than the Astros are to not hit into one. Yu Darvish is one of the most likely pitchers to throw a no-hitter, based on batting average against, and the Astros are one of the most-likely teams to be no-hit, according to the lineup's batting averages. That's probably why a Rangers vs. Astros matchups featuring Yu Darvish on the mound turned into narrow brushes with history - twice.

* * *

On April 2, 2013, the second day of the season, Yu Darvish very nearly pitched the 24th perfect game in the history of Major League Baseball. The following table shows the starting lineup from that day with their career averages against Yu Darvish.

Position	Player	Career AVGYD
C	Justin Maxwell	0.333
1B	Carlos Pena	0.000
2B	Jose Altuve	0.133
3B	Brett Wallace	0.167
SS	Marwin Gonzalez	0.333
LF	Chris Carter	0.000
CF	Justin Maxwell	0.333
RF	Rick Ankiel	0.000
DH	Jason Castro	0.167
Source: Baseball-Reference.com

The Astros lineup on April 2 included three players who still have never gotten a hit off of Darvish. Despite the Astros and the Rangers playing in the same division, most Astros do not have many plate attempts against Darvish because they're all young (the batters and Yu Darvish) and just switched to the AL, so they simply haven't played him often. The career batting averages shown above are subject to small sample size bias. Knowing what we know about the starting lineup today, which includes an extra handful of starts against Darvish from the 2013 season, the Astros had a 0.569% chance of being no-hit that day. On average, one of every 176 games between this lineup and Yu Darvish would produce a no-hitter.

The lineup shown initially includes four batters who have never gotten a hit off of Darvish, one player who has never faced Darvish, and only one batter hitting above .167 against Darvish (Matt Dominguez, .273). That lineup is even more likely to be no-hit by Yu Darvish than the one that Darvish actually successfully no-hit for 26 batters. When the Astros match up against Darvish in 2014, pay attention: it's more likely that Yu Darvish will no-hit the Astros than it is that the Astros will naturally not record a hit against the average pitcher and that Yu Darvish will pitch a no-hitter against a the average lineup.

And the probability that Marwin Gonzalez would get a hit in a randomly selected at-bat against Yu Darvish, including one that happened after 26 consecutive outs on April 2, 2013? That probability is equal to his career batting average against Darvish: 33.3%.